Toowoomba Flash Flooding

What was the probability of this flash flood occurring?

Did ENSO affect the probability of this flash flood occurring?

1. Quality Assurance

1. Quality Assurance


2. Non-stationarity of Extremes

1. Quality Assurance


2. Non-stationarity of Extremes


3. Extremes in Continuous Space with Dependence

1. Quality Assurance


2. Non-stationarity of Extremes


3. Extremes in Continuous Space with Dependence


4. Extremal Dependence

Let \(X_i\) be a sequence of iid random variables, define \[M_{n} = \max\{X_1, \dots, X_{n}\}.\]

The distribution function of \(M_n\) is \[\mathbb{P}(M_n \leq z) = \mathbb{P}(X_1 \leq z, \dots, X_n \leq z) = \mathbb{P}(X \leq z)^n = F(z)^n,\] where \(F(z)\) is the distribution function of \(X\) .

Let \(z^F\) denote the right endpoint of the support of \(F\) , \[z^F = \sup \{z : F(z) < 1\},\] then as \(n \rightarrow \infty\), \(F(z)^n \rightarrow 0\) for any \(z < z^F\).

If there exists sequences of constants \(\{a_n\} > 0\) and \(\{b_n\} \in \mathbb{R}\) such that \[\mathbb{P} \left\{\dfrac{M_n - b_n}{a_n} \leq z \right \} \rightarrow G(z) \quad\hbox{as}\quad\, n \rightarrow \infty \] where \(G(z)\) is a non-degenerate distribution function, then \(G(z)\) is a member of the generalised extreme value (GEV) family \[ G(z) = \exp \left\{ - \left[ 1 + \xi \left(\dfrac{z-\mu}{\sigma}\right) \right]_+^{-1 / \xi} \right\},\] where \([v]_+ = \max \left\lbrace 0,v \right\rbrace\), \(\mu \in \mathbb{R}\), \(\sigma \in \mathbb{R}^+\) and \(\xi \in \mathbb{R}\).

(Fisher and Tippett 1928, Gnendenko 1943)

Why approximate the \(\mathbb{P}(M_n \leq z)\) by the GEV distribution?

Why approximate the \(\mathbb{P}(M_n \leq z)\) by the GEV distribution?




Rainfall observations aren't independent

Why approximate the \(\mathbb{P}(M_n \leq z)\) by the GEV distribution?





Rainfall observations aren't independent




Rainfall observations aren't identically distributed

For the GEV: \(\mu\) is the location parameter, \(\sigma\) is the scale parameter, \(\xi\) is the shape parameter.

GEV Parameters (linear functions): \[ \mu = l_{\mu}(\hbox{geographic covariates, climate covariates})\] \[\sigma = l_{\sigma}(\hbox{geographic covariates, climate covariates})\] \[\xi = \hbox{constant}\]

Use Southern Oscillation Index (SOI) as a measure for ENSO strength.

• Extremes in continuous space with dependence
  
  
• Natural extension from univariate extreme value theory
  
  
• Univariate marginal distributions are GEV distributions
  
  
• Can simulate from these processes

Let \(\{Z_i\}_{i \geq 1}\) be a sequence of independent copies of a stochastic process \(\{ Z(x) : x \in \mathcal{X} \subset \mathbb{R}^2 \}\).

The process \(Z(x)\) is max-stable, if there exist normalising functions, \(\{a_n(x)\} \in \mathbb{R}^+\) and \(\{b_n(x)\} \in \mathbb{R}\), such that \[ Z(x) \stackrel{d}{=} \lim_{n\to\infty} \dfrac{ \max _{i=1,\dots,n} Z_i(x) - b_{n}(x) }{ a_{n}(x) }, \quad x \in \mathcal{X}.\] If the limiting process for the partial maxima process exists and is non-degenerate, then it is a max-stable process.

(De Haan 2006)

Any non-degenerate simple max-stable process \(\{ Z(x): x \in \mathcal{X}\}\) defined on a compact set \(\mathcal{X} \subset \mathbb{R}^2\), with continuous sample paths satisfies \[ Z(x) \stackrel{d}{=} \max_{i \geq 1} \zeta_i Y_i(x), \quad\quad x \in \mathcal{X}, \] where \(\{\zeta_i: i \geq 1 \}\) are points of a Poisson process on \((0, \infty)\) with intensity \(\zeta^{-2}\hbox{d}\zeta\), and \(Y_i\) are independent copies of a non-negative stochastic process \(\{Y(x): x\in \mathcal{X}\}\) with continuous sample paths such that the \(\mathbb{E}\lbrace Y(x) \rbrace = 1\) for all \(x \in \mathcal{X}\).

(De Haan 1984, Schlather 2002)

Was the Toowoomba flash flood more likely due to the 2010-2011 La Nina ?

Yes ~ 85% more likely compared with a El Nino year.

If we had a similar strength La Nina, what is the probability of this flash flood occurring?

1 in 7.5 (0.134)

(For details see Saunders et al. 2017.)

Can we do this type of modelling for all of Australia?

Form clusters based on extremal dependence!
(Bernard et al 2013)

• Use only the raw annual maxima
• No information about climate or topography

Use the F-madogram distance (Cooley et al 2006) \[d(x_i, x_j) = \tfrac{1}{2} \mathbb{E} \left[ \left| F_i(M_{x_i}) - F_j(M_{x_j})) \right| \right]\] where \(M_{x_i}\) is the annual maximum rainfall at location \(x_i \in \mathbb{R}^2\) and \(F_i\) is the distribution function of \(M_{x_i}\).

This distance can be estimated non-parametrically.

For \(M_{x_i}\) and \(M_{x_j}\) with common GEV marginals, \(\theta(x_i - x_j)\) is \[\mathbb{P}\left( M_{x_i} \leq z, M_{x_j} \leq z \right) = \left[\mathbb{P}(M_{x_i}\leq z)\mathbb{P}(M_{x_i}\leq z)) \right]^{\tfrac{1}{2}\theta(x_i - x_j)}. %= \exp\left(\dfrac{-\theta(h)}{z}\right),\]

The range of \(\theta(x_i - x_j)\) is \([1 , 2]\).

Can write our distance measure as a function of the extremal coefficient, \(\theta(x_i - x_j)\), \[d(x_i, x_j) = \dfrac{\theta(x_i - x_j) - 1}{2(\theta(x_i - x_j) + 1)}.\]

Therefore the range of \(d(x_i, x_j)\) is \([0 , 1/6]\).

Partitioning around Medoids (PAM): (Kaufman and Rousseeuw 1990)

1. Randomly select an initial set of \(K\) stations. These are the set of the initial medoids.
2. Assign each station, \(x_i\), to its closest medoid, \(m_k\), based on the F-madogram distance.
3. For each cluster, \(C_k\), update the medoid according to \[m_k = \mathop{\mathrm{argmin}}\limits_{x_i \in C_k} \sum_{x_j \in C_k} d(x_i, x_j).\]
4. Repeat steps 2. – 4. until the medoids are no longer updated.

Let \(d_e(x_i,x_j)\) be the Euclidean distance between x and y.
Consider the \(\max\{d_e(x_i,x_j), 2\}\) as the clustering distance.

1. Each station starts in its own cluster
2. For each pair of clusters, \(C_k\) and \(C_k'\), define the distance between the clusters as \[d(C_k, C_{k'}) = \frac{1}{|C_k| |C_{k'}|} \sum_{x_k \in C_k} \sum_{x_{k'} \in C_{k'}} d(x_k, x_{k'}).\]
3. Merge the the clusters with the smallest distance
4. Update the distances relative to the new cluster
5. Repate steps 3 - 5, until all points are combined in a single cluster

• Classify a station relative to its closest neighbours
• Use a weighted classification \(w\)-kNN

Where can we assume a common dependence structure?

• Max-stable models powerful tool for modelling extremes

• Dependence of annual maxima in Australia is highly variable and highly localised

• Exercise caution in our modelling assumptions

e. katerobinsonsaunders@gmail.com

t. @katerobsau

g. github.com/katerobsau